What is the coefficient of n² in a quadratic sequence determined from its differences?

In a quadratic sequence, the second difference is constant, and it provides a crucial piece of information about the coefficient of the n² term in the quadratic expression that defines the sequence. When you take the first differences of a quadratic sequence, you get a new sequence that represents how the original values are changing. When you then find the differences of this new sequence (the second differences), this constant value is directly related to the coefficient of the n² term in the quadratic equation.

Specifically, if the second difference is 'd', then the coefficient of n² in the quadratic formula, which typically takes the form an² + bn + c, is equal to half of that second difference. Thus, if the second difference is constant throughout the sequence, dividing that constant by 2 yields the coefficient that captures the quadratic growth rate in the original sequence.

This relationship helps connect the nature of the sequence to its mathematical representation, making it essential for identifying the underlying structure of quadratic sequences.